Abstract

The multidimensional nonlinear Schrödinger equation governs the spatial and temporal evolution of an optical field inside a nonlinear dispersive medium. Although spatial (diffractive) and temporal (dispersive) effects can be studied independently in a linear medium, they become mutually coupled in a nonlinear medium. We present the results of numerical simulations showing this spatiotemporal coupling for ultrashort pulses propagating in dispersive Kerr media. We investigate how spatiotemporal coupling affects the behavior of the optical field in each of the four regimes defined by the type of group-velocity dispersion (normal or anomalous) and the type of nonlinearity (focusing or defocusing). We show that dispersion, through spatiotemporal coupling, can either enhance or suppress self-focusing and self-defocusing. Similarly, we demonstrate that diffraction can either enhance or suppress pulse compression or broadening. We also discuss how these effects can be controlled with optical phase modulation, such as that provided by a lens (spatial phase modulation) or frequency chirping (temporal phase modulation).

© 1995 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  2. G. I. Stegeman, "Nonlinear guided wave optics," in Contemporary Nonlinear Optics, G. P. Agrawal and R. W. Boyd, eds. (Academic, San Diego, Calif., 1992), Chap. 1, pp. 1–40.
    [CrossRef]
  3. D. Mihalache, M. Bertoletti, and C. Sibilia, "Nonlinear wave propagation in planar structures," in Progress in Optics, E Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, chap. 4, pp. 252–288.
  4. A. T. Ryan and G. P. Agrawal, "Pulse compression and spatial phase modulation in normally dispersive nonlinear Kerr media," Opt. Lett. 20, 306 (1995).
    [CrossRef] [PubMed]
  5. A. T. Ryan and G. P. Agrawal, "Dispersion-induced beam narrowing in a self-defocusing medium," in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTHD6.
  6. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086 (1992).
    [CrossRef]
  7. I. Androsch and P. Glas, "Influence of spatio-temporal refractive index changes in a GaAs étalon on the generation of ultrashort pulses in a Nd-phosphate glass laser," Opt. Commun. 105, 126 (1994).
    [CrossRef]
  8. G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Use of ZnS as a self-focusing element in a self-starting Kerr lens modelocked Ti:sapphire laser," in Generation, Amplification, and Measurement of Ultrashort Laser Pulses, R. B. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 76 (1994).
    [CrossRef]
  9. R. W. Ziolkowski and J. B. Judkins, "Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time," J. Opt. Soc. Am. B 10, 186 (1993).
    [CrossRef]
  10. J. E. Rothenberg, "Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses," Opt. Lett. 17, 1340 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. G. R. Hadley, "Wide-angle beam propagation using Padé approximation operators," Opt. Lett. 17, 1426 (1992); "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743 (1992).
    [CrossRef] [PubMed]
  13. X. H. Wang and G. K. Cambrell, "Simulation of strong nonlinear effects in optical waveguides," J. Opt. Soc. Am. B 10, 2048 (1993).
    [CrossRef]
  14. L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
    [CrossRef] [PubMed]
  15. V. E. Zakharov and A. B. Shabat, "Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 84, 62 (1972).
  16. X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, "Self-focusing of chirped optical pulses in nonlinear dispersive media," Phys. Rev. A 49, 4085 (1994).
    [CrossRef] [PubMed]
  17. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  18. D. Strickland and P. B. Corkum, "Resistance of short pulses to self-focusing," J. Opt. Soc. Am. B 11, 492 (1994).
    [CrossRef]
  19. A. Dreischuh, E. Eugenieva, and S. Dinev, "Pulse shaping and shortening by spatial filtering of an induced-phase-modulated probe wave," IEEE J. Quantum Electron. 30, 1656 (1994).
    [CrossRef]

1995 (1)

1994 (4)

X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, "Self-focusing of chirped optical pulses in nonlinear dispersive media," Phys. Rev. A 49, 4085 (1994).
[CrossRef] [PubMed]

A. Dreischuh, E. Eugenieva, and S. Dinev, "Pulse shaping and shortening by spatial filtering of an induced-phase-modulated probe wave," IEEE J. Quantum Electron. 30, 1656 (1994).
[CrossRef]

I. Androsch and P. Glas, "Influence of spatio-temporal refractive index changes in a GaAs étalon on the generation of ultrashort pulses in a Nd-phosphate glass laser," Opt. Commun. 105, 126 (1994).
[CrossRef]

D. Strickland and P. B. Corkum, "Resistance of short pulses to self-focusing," J. Opt. Soc. Am. B 11, 492 (1994).
[CrossRef]

1993 (3)

1992 (4)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

J. E. Rothenberg, "Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses," Opt. Lett. 17, 1340 (1992).
[CrossRef] [PubMed]

G. R. Hadley, "Wide-angle beam propagation using Padé approximation operators," Opt. Lett. 17, 1426 (1992); "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743 (1992).
[CrossRef] [PubMed]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

1990 (1)

1972 (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 84, 62 (1972).

Agrawal, G. P.

A. T. Ryan and G. P. Agrawal, "Pulse compression and spatial phase modulation in normally dispersive nonlinear Kerr media," Opt. Lett. 20, 306 (1995).
[CrossRef] [PubMed]

X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, "Self-focusing of chirped optical pulses in nonlinear dispersive media," Phys. Rev. A 49, 4085 (1994).
[CrossRef] [PubMed]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

A. T. Ryan and G. P. Agrawal, "Dispersion-induced beam narrowing in a self-defocusing medium," in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTHD6.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

Akhmediev, N.

Androsch, I.

I. Androsch and P. Glas, "Influence of spatio-temporal refractive index changes in a GaAs étalon on the generation of ultrashort pulses in a Nd-phosphate glass laser," Opt. Commun. 105, 126 (1994).
[CrossRef]

Ankiewicz, A.

Bertoletti, M.

D. Mihalache, M. Bertoletti, and C. Sibilia, "Nonlinear wave propagation in planar structures," in Progress in Optics, E Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, chap. 4, pp. 252–288.

Cambrell, G. K.

Cao, X. D.

X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, "Self-focusing of chirped optical pulses in nonlinear dispersive media," Phys. Rev. A 49, 4085 (1994).
[CrossRef] [PubMed]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

Corkum, P. B.

Dinev, S.

A. Dreischuh, E. Eugenieva, and S. Dinev, "Pulse shaping and shortening by spatial filtering of an induced-phase-modulated probe wave," IEEE J. Quantum Electron. 30, 1656 (1994).
[CrossRef]

Dreischuh, A.

A. Dreischuh, E. Eugenieva, and S. Dinev, "Pulse shaping and shortening by spatial filtering of an induced-phase-modulated probe wave," IEEE J. Quantum Electron. 30, 1656 (1994).
[CrossRef]

Eugenieva, E.

A. Dreischuh, E. Eugenieva, and S. Dinev, "Pulse shaping and shortening by spatial filtering of an induced-phase-modulated probe wave," IEEE J. Quantum Electron. 30, 1656 (1994).
[CrossRef]

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

Glas, P.

I. Androsch and P. Glas, "Influence of spatio-temporal refractive index changes in a GaAs étalon on the generation of ultrashort pulses in a Nd-phosphate glass laser," Opt. Commun. 105, 126 (1994).
[CrossRef]

Hadley, G. R.

Haus, H. A.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

Judkins, J. B.

Krasinski, J. S.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Use of ZnS as a self-focusing element in a self-starting Kerr lens modelocked Ti:sapphire laser," in Generation, Amplification, and Measurement of Ultrashort Laser Pulses, R. B. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 76 (1994).
[CrossRef]

Liou, L. W.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

McKinstrie, C. J.

X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, "Self-focusing of chirped optical pulses in nonlinear dispersive media," Phys. Rev. A 49, 4085 (1994).
[CrossRef] [PubMed]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

Mihalache, D.

D. Mihalache, M. Bertoletti, and C. Sibilia, "Nonlinear wave propagation in planar structures," in Progress in Optics, E Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, chap. 4, pp. 252–288.

Pearson, G. W.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Use of ZnS as a self-focusing element in a self-starting Kerr lens modelocked Ti:sapphire laser," in Generation, Amplification, and Measurement of Ultrashort Laser Pulses, R. B. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 76 (1994).
[CrossRef]

Radzewicz, C.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Use of ZnS as a self-focusing element in a self-starting Kerr lens modelocked Ti:sapphire laser," in Generation, Amplification, and Measurement of Ultrashort Laser Pulses, R. B. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 76 (1994).
[CrossRef]

Rothenberg, J. E.

Ryan, A. T.

A. T. Ryan and G. P. Agrawal, "Pulse compression and spatial phase modulation in normally dispersive nonlinear Kerr media," Opt. Lett. 20, 306 (1995).
[CrossRef] [PubMed]

A. T. Ryan and G. P. Agrawal, "Dispersion-induced beam narrowing in a self-defocusing medium," in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTHD6.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 84, 62 (1972).

Sibilia, C.

D. Mihalache, M. Bertoletti, and C. Sibilia, "Nonlinear wave propagation in planar structures," in Progress in Optics, E Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, chap. 4, pp. 252–288.

Silberberg, Y.

Soto-Crespo, J. M.

Stegeman, G. I.

G. I. Stegeman, "Nonlinear guided wave optics," in Contemporary Nonlinear Optics, G. P. Agrawal and R. W. Boyd, eds. (Academic, San Diego, Calif., 1992), Chap. 1, pp. 1–40.
[CrossRef]

Strickland, D.

Wang, X. H.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 84, 62 (1972).

Ziolkowski, R. W.

IEEE J. Quantum Electron. (2)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

A. Dreischuh, E. Eugenieva, and S. Dinev, "Pulse shaping and shortening by spatial filtering of an induced-phase-modulated probe wave," IEEE J. Quantum Electron. 30, 1656 (1994).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

I. Androsch and P. Glas, "Influence of spatio-temporal refractive index changes in a GaAs étalon on the generation of ultrashort pulses in a Nd-phosphate glass laser," Opt. Commun. 105, 126 (1994).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (2)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, "Spatiotemporal instabilities in dispersive nonlinear media," Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, "Self-focusing of chirped optical pulses in nonlinear dispersive media," Phys. Rev. A 49, 4085 (1994).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 84, 62 (1972).

Other (5)

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Use of ZnS as a self-focusing element in a self-starting Kerr lens modelocked Ti:sapphire laser," in Generation, Amplification, and Measurement of Ultrashort Laser Pulses, R. B. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 76 (1994).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

G. I. Stegeman, "Nonlinear guided wave optics," in Contemporary Nonlinear Optics, G. P. Agrawal and R. W. Boyd, eds. (Academic, San Diego, Calif., 1992), Chap. 1, pp. 1–40.
[CrossRef]

D. Mihalache, M. Bertoletti, and C. Sibilia, "Nonlinear wave propagation in planar structures," in Progress in Optics, E Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, chap. 4, pp. 252–288.

A. T. Ryan and G. P. Agrawal, "Dispersion-induced beam narrowing in a self-defocusing medium," in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTHD6.

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Figures (9)

Fig. 1
Fig. 1

Results of one-dimensional simulations showing the two types of behavior described by the NSE for a self-focusing nonlinearity. For an input Gaussian field with N = 3 we have (a) the evolution through compression, splitting, and recovery of a bright spatial soliton and (b) the monotonic broadening of a pulse in the normal-dispersion regime.

Fig. 2
Fig. 2

To show the influence of spatial phase modulation on the wave collapse that occurs for N = 3 and d = −1, the integrated pulse and beam widths as a function of propagation distance are plotted. In (a) the pulse compression can be increased or decreased depending on the type of lens employed, and in (b) the beam width behaves in a corresponding fashion.

Fig. 3
Fig. 3

The amount of localized pulse compression in the normal-dispersion regime depends on both the amount of dispersion and the strength of the nonlinearity. The normalized pulse width at ξ = 0 is plotted, showing that (a) the amount of localized compression decreases as the strength of the dispersion is increased and that (b) for d = 1, compression of the pulse below its input width requires a strong nonlinearity (N = 5).

Fig. 4
Fig. 4

Interplay of spatial self-focusing and dispersive pulse broadening. In (a), for |N2/d| = 16 the integrated pulse width as a function of dispersion length, the stronger self-focusing (for d = 1) eventually creates a broader pulse. In (b) for N = 3, increasing the strength of the dispersion first enhances (d = 0.5) the self-focusing and then saturates it so that for d = 1 the self-focusing is actually weaker.

Fig. 5
Fig. 5

Influence of spatial phase modulation on the temporal behavior for d = 0.5 and two different nonlinearites. In (a) the plot of the pulse width at ξ = 0 shows that, depending on the lens, the localized compression can be either enhanced or suppressed with spatial phase modulation, whereas in (b) the localized compression is seen to enhance the integrated-pulse broadening.

Fig. 6
Fig. 6

In the anomalous-dispersion regime with a self-defocusing nonlinearity the pulse and the beam broaden monotonically. In (a) including diffraction in the model is seen initially to slightly increase the pulse width at the beam center, whereas in (b) the effect of diffraction on the spatially integrated pulse width is the opposite: the rate of broadening is decreased.

Fig. 7
Fig. 7

In the normal-dispersion regime with a self-defocusing nonlinearity the pulse compresses and the beam broadens. In (a) the effect of including diffractive effects on the spatially integrated pulse is to reduce the broadening, and in (b) spatial phase modulation is employed to enhance the pulse compression.

Fig. 8
Fig. 8

To show the effect of the N and d parameters, the width (FWHM) of the spatial intensity distribution through the center of the pulse (τ = 0) normalized to its input value is plotted as a function of propagation distance. In (a) the effect of increasing the strength of the nonlinearity is to increase the localized spatial narrowing, whereas in (b) the dispersion-induced enhancement of the localized narrowing saturates near d = 5.

Fig. 9
Fig. 9

Temporal phase modulation can be used to control the beam width in a self-defocusing medium. In (a) an upchirp (C > 0) is seen to reduce the localized beam narrowing that occurs at τ = 0, whereas a downchirp (C < 0) is seen to enhance it. In (b) the effect of chirp on the time-integrated beam width is such that an upchirp leads to a slower beam broadening and a downchirp increases the beam width. In an anomalously dispersive medium the effects of the upchirp and the downchirp would be reversed.

Equations (4)

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i u ζ + 1 2 2 u ξ 2 - d 2 2 u τ 2 + sgn ( n 2 ) N 2 u 2 u = 0.
u ( ξ , τ , 0 ) = exp ( - ξ 2 2 - τ 2 2 ) exp [ i ϕ ( ξ , τ ) ] .
i u ζ + 1 2 2 u ξ 2 + sgn ( n 2 ) N 2 u 2 u = 0.
i u ζ - sgn ( β 2 ) 2 2 u τ 2 + sgn ( n 2 ) N 2 u 2 u = 0.

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